Morley Company is attempting to determine the best location for a new machine in an existing layout of three machines. The existing machines are located at the following x1, x2 coordinates on the shop floor:

Machine 1: x1 = 1, x2 = 7

Machine 2:x1 = 5, x2 = 9

Machine 3:x1 = 6, x2 = 2

a. Develop a goal programming model that can be solved to minimize the total distance of the new machine from the three existing machines. The distance is to be measured rectangularly. For example, if the location of the new machine is (x1 = 3, x2 = 5), it is considered to be a distance of |3 – 1| + |5 – 7| = 2 + 2 = 4 from machine 1.

x1 = first coordinate of the new machine location

x2 = second coordinate of the new machine location

d+i = amount by which the x1 coordinate of the new machine exceeds the x1 coordinate of machine i (i = 1, 2, 3)

d–i = amount by which the x1 coordinate of machine i exceeds the x1 coordinate of the new machine (i = 1, 2, 3)

e+i = amount by which the x2 coordinate of the new machine exceeds the x2 coordinate of machine i (i = 1, 2, 3)

e–i = amount by which the x2 coordinate of machine i exceeds the x2 coordinate of the new machine (i = 1, 2, 3)

b. What is the optimal location for the new machine?